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Theorem grlimprop2 47953
Description: Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.)
Hypotheses
Ref Expression
grlimprop.v 𝑉 = (Vtx‘𝐺)
grlimprop.w 𝑊 = (Vtx‘𝐻)
grlimprop2.n 𝑁 = (𝐺 ClNeighbVtx 𝑣)
grlimprop2.m 𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))
grlimprop2.i 𝐼 = (iEdg‘𝐺)
grlimprop2.j 𝐽 = (iEdg‘𝐻)
grlimprop2.k 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}
grlimprop2.l 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}
Assertion
Ref Expression
grlimprop2 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
Distinct variable groups:   𝑣,𝐹   𝑣,𝐺   𝑣,𝐻   𝑣,𝑉   𝑓,𝐹,𝑔,𝑣   𝑓,𝐺,𝑔,𝑖,𝑥   𝑓,𝐻,𝑔,𝑖,𝑥   𝑥,𝐼   𝑥,𝐽   𝑖,𝐾   𝑖,𝐿   𝑓,𝑀,𝑔,𝑖,𝑥   𝑓,𝑁,𝑔,𝑖,𝑥   𝑣,𝑖
Allowed substitution hints:   𝐹(𝑥,𝑖)   𝐼(𝑣,𝑓,𝑔,𝑖)   𝐽(𝑣,𝑓,𝑔,𝑖)   𝐾(𝑥,𝑣,𝑓,𝑔)   𝐿(𝑥,𝑣,𝑓,𝑔)   𝑀(𝑣)   𝑁(𝑣)   𝑉(𝑥,𝑓,𝑔,𝑖)   𝑊(𝑥,𝑣,𝑓,𝑔,𝑖)

Proof of Theorem grlimprop2
StepHypRef Expression
1 grlimdmrel 47947 . . . . 5 Rel dom GraphLocIso
21ovrcl 7472 . . . 4 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V))
3 id 22 . . . 4 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
4 df-3an 1089 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) ↔ ((𝐺 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)))
52, 3, 4sylanbrc 583 . . 3 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)))
6 grlimprop.v . . . 4 𝑉 = (Vtx‘𝐺)
7 grlimprop.w . . . 4 𝑊 = (Vtx‘𝐻)
8 grlimprop2.n . . . 4 𝑁 = (𝐺 ClNeighbVtx 𝑣)
9 grlimprop2.m . . . 4 𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))
10 grlimprop2.i . . . 4 𝐼 = (iEdg‘𝐺)
11 grlimprop2.j . . . 4 𝐽 = (iEdg‘𝐻)
12 grlimprop2.k . . . 4 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}
13 grlimprop2.l . . . 4 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}
146, 7, 8, 9, 10, 11, 12, 13isgrlim2 47950 . . 3 ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
155, 14syl 17 . 2 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
1615ibi 267 1 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951  dom cdm 5685  cima 5688  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014   ClNeighbVtx cclnbgr 47805   GraphLocIso cgrlim 47943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-vtx 29015  df-iedg 29016  df-clnbgr 47806  df-isubgr 47847  df-grim 47864  df-gric 47867  df-grlim 47945
This theorem is referenced by: (None)
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